scholarly journals Torsion of rational elliptic curves over cubic fields

2016 ◽  
Vol 46 (6) ◽  
pp. 1899-1917 ◽  
Author(s):  
Enrique González-Jiménez ◽  
Filip Najman ◽  
José M. Tornero
Keyword(s):  
Elliptic Curves ◽  
Cubic Fields

Elliptic curves with good reduction everywhere over cubic fields

2015 ◽  
Vol 11 (04) ◽  
pp. 1149-1164 ◽  
Author(s):  
Nao Takeshi
Keyword(s):  
Elliptic Curves ◽  
Infinite Family ◽  
Good Reduction ◽  
Cubic Fields

We give a criterion for cubic fields over which there exist no elliptic curves with good reduction everywhere, and we construct a certain infinite family of cubic fields over which there exist elliptic curves with good reduction everywhere.

scholarly journals The Selmer groups and the ambiguous ideal class groups of cubic fields

1996 ◽  
Vol 54 (2) ◽  
pp. 267-274
Author(s):  
Yen-Mei J. Chen
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Upper Bound ◽  
Selmer Groups ◽  
Ideal Class ◽  
Selmer Group ◽  
Class Groups ◽  
Ideal Class Groups ◽  
Cubic Fields ◽  
Rational Isogeny

In this paper, we study a family of elliptic curves with CM by which also admits a ℚ-rational isogeny of degree 3. We find a relation between the Selmer groups of the elliptic curves and the ambiguous ideal class groups of certain cubic fields. We also find some bounds for the dimension of the 3-Selmer group over ℚ, whose upper bound is also an upper bound of the rank of the elliptic curve.

scholarly journals Torsion of elliptic curves over cyclic cubic fields

2018 ◽  
Vol 88 (319) ◽  
pp. 2443-2459 ◽  
Author(s):  
Maarten Derickx ◽  
Filip Najman
Keyword(s):  
Elliptic Curves ◽  
Cubic Fields ◽  
Cyclic Cubic Fields

Some remarks on a theorem of Parent and generalizing Ogg's conjecture

2009 ◽  
Vol 147 (2) ◽  
pp. 285-293
Author(s):  
S. KAMIENNY
Keyword(s):  
Elliptic Curves ◽  
Number Fields ◽  
Cubic Fields ◽  
Weil Group

AbstractWe carry out an Eisenstein prime descent to prove the finiteness of the Mordell-Weil group of the Eisenstein quotients ofJ1(p) for certain values ofpthat are relevant to torsion in elliptic curves over cubic fields. We then use this to recover some results of Parent. Our methods suggest a possible generalization of Ogg's conjecture for torsion in elliptic curves over number fields.

scholarly journals The $2$-class groups of cubic fields and $2$-descents on elliptic curves

1992 ◽  
Vol 44 (4) ◽  
pp. 557-565 ◽  
Author(s):  
Mayumi Kawachi ◽  
Shin Nakano
Keyword(s):  
Elliptic Curves ◽  
Class Groups ◽  
Cubic Fields
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